Representations of conditional means (Q2761600)

A unified treatment of integral representations of conditional expectations is given in terms of Bochner integrals. In particular, for Gaussian processes, the following representation theorem is derived: Let \((Y_t)_{t\geq 0}\) be a real continuous Gaussian martingale and \(X\) a further Gaussian random variable such that for all \(t_1,\dots,t_n\), \((X,Y_{t_1}, \dots,Y_{t_n})\) is Gaussian. Then there is a deterministic kernel \(K\) such that for all \(t\geq 0\), NEWLINE\[NEWLINEE(X\mid Y_s,s\leq t)= E(X\mid Y_0)+ \int^t_0 K(t,s)dY_s.NEWLINE\]